Abstract
The research in fracture mechanics of concrete is reviewed in an attempt to reconcile the different, and sometimes contradictory, approaches for evaluation of the fracture energy, Gf. An improved method to measure Gf for beams in three point bending is presented. This method is expected to provide a more reliable characterization of the fracture toughness of concrete.
Similar content being viewed by others
References
Griffith, A.A., ‘The Phenomena of Rupture and Flow in Solids’,Phil. Trans R. Soc., Series A221 (1920) pp. 163–198.
Kaplan, M.F., ‘Crack Propagation and the Fracture of Concrete’,Am. Concrete Inst. J. 58, 11 (1961) 591–610.
Romualdi, J.P., Batson, G.B., ‘Mechanics Crack Arrest in Concrete’,J. Eng. Mech. Div. ASCE, (1963) pp. 147–167.
Glucklich, J., ‘Fracture of Plain concrete’,Journal of the Engineering Mechanics Division, ASCE 89, EM6 (1963) 127–138.
Hughes, B.P., Chapman, G.P., ‘The complete Stress Strain Curve for Concrete in Direct Tension’, RILEM Bulletin, No. 30, pp. 95–97 March 1966.
Mindess, S., ‘Fracture Toughness Testing of Cement and Concrete’, in Fracture Mechanics of Concrete edited by Carpinteri, A. and Ingraffea, A.R. (Martinus Nijohff Publishers, Dordrecht, The Netherlands, 1984) pp. 67–110.
Carpinteri, A., ‘Application of Fracture Mechanics to Concrete Structures’, ASCE,Journal of the Structural Division 108, ST4 (1982) pp. 833–848.
ACI Committee 224, ‘Control of Cracking in Concrete Structures’, Report No. ACI 224R-80,Concrete International,2, (10) (1980).
Gustafsson, P. J., ‘Fracture Mechanics Studies of Non Yielding Materials like Concrete: Modeling of Tensile Fracture and Applied Strength Analyses’, Report TVBM-1007, Division of Building Materials, Lund, Sweden, 1985.
RILEM Technical Committee 50-FMC, Proposed Recommendation, ‘Determination of the Fracture Energy of Mortar and Concrete by Means of Three Point Bend Tests on Notched Beams’, Jan 1982, Revised June 1982.
RILEM Technical Committee 50-FMC, Draft Recommendation, ‘Determination of the Fracture Energy of Mortar and Concrete by Means of Three-Point Bend Tests on Notched Beams,Mater. Struct. 18 (106) (1985) 285–290.
Hillerborg, A., ‘Results of Three Comparative Test Series for Determining the Fracture Energy of the Concrete’,Mater. Struct. 18 (107) (1985).
Idem, Concrete Fracture Energy Tests Performed by nine laboratories According to a Draft RILEM Recommendation, Report to RILEM TC50-FMC, Report TVBM-3015, Division of Building Materials, Lund Institute of Technology, S-221 00 Lund, Sweden, 1983.
Idem, Additional Concrete Fracture Energy Tests Performed by 6 Laboratories According to a Draft RILEM Recommendation, Report TVBM-3017, Division of Building Materials, Lund Institute of Technology, S-221 00 Lund, Sweden, 1984.
Petersson, P. E., ‘Fracture Energy of Concrete: Method of Determination, Practical Performance and Experimental Results’,Cem. Concr. Res,10 (1980) 79–101.
“Somayaji, S., »A Discussion of the Papers: Fracture Energy of Concrete: Method of Determination, Practical Performance and Experimental Results’, by P.E. Petersson”,Cem. Concr. Res.,10 (1980) 471–474.
Hillerborg, A., ‘The Theoretical Basis of a Method to Determine the Fracture Energy of Concrete’,Mater. Struct,18 (106) (1985) 291–296.
Gustafsson, P.J., Hillerborg, A., ‘Improvements in Concrete Design Achieved Through Application of Fracture Mechanics’, Application of Fracture Mechanics to Cementitious Composites, NATO-ARW, Sept 4–7, 1984, Northwestern University (edited by S.P. Shah) pp. 487–500.
Swartz, S.E., Siew, H.C., ‘Load Versus Strain Control in Fracture Energy Measurements for Concrete’, Proc. 1986 SEM Spring Conference on Experimental Mechanics, Society for Experimental Mechanics, New Orleans, June 1986, pp. 16–22.
Cook, D.J., Crookham, G., ‘A Discussion of the Paper: Notch Sensitivity and Fracture Toughness of Concrete’,Cem. Concr. Res. 8 (1978) 387–388.
Brown, J.H., ‘Measuring the Fracture Toughness of Cement Paste and Mortar’,Magazine of Concrete Research,24 No. 81 (1972) 185–196.
Swartz, S.E., Rood, S.M., ‘Fracture Toughness Testing of Concrete Beams in Three-Point Bending: Phase I, Small Beams’, Proc., 1985 SEM Spring Conference on Experimental Mechanics, Society for Experimental Mechanics, Las Vegas, pp. 119–126.
Nakayama, J., ‘Direct Measurement of Fracture Energies of Brittle Heterogeneous Materials’,J. Amer. Ceram. Soc. 48 (9) (1965) 583–587.
Davidge, R.W., Tappin, G., ‘The Effective Surface Energy of Brittle Materials’,J. Mater. Sci.,3 (1968) 165–173.
Wecharatana, M., Swartz, S.E., Review of Current Notations on Fracture Toughness Testing of Concrete and Rock, Report to the Committee on Fracture Mechanics of Concrete and Rock, Society of Experimental Mechanics, Las Vegas, June 1985.
Hillerborg, A., ‘Analysis of One Single Crack’,Fracture Mechanics of Concrete (edited by F. H. Wittman) (Elsevier Science Publishers, Amsterdam, 1983) pp. 223–249.
Hillerborg, A., Modeer, M., Petersson, P. E., ‘Analysis of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements’,Cem. Concr. Res. 6 (1976) 773–782.
Go, C. G., Swartz, S. E., ‘Energy Methods for Fracture Toughness Determination in Concrete’, Proc., V Int. Congress on Experimental Mechanics, Montreal, Canada, 1984, pp. 453–459. Also in: Experimental Mechanics26 (3) (1986) 292–298.
Turner, C. E., ‘Fracture Toughness and Specific Fracture Energy: A Re-Analysis of Results’,Mater. Sci. Eng.,11 (1973) 275–282.
Plati, E., Williams, J.G., ‘The Determination of the Fracture Parameters for Polymers in Impact’,Polymer Eng. Sci. 15 (1975) 470–477.
Rice, J.R., Paris, P.C., Merkle, J.G., ‘Some Further Aspects of J-Integral Analysis and Estimates’,Progress in Flaw Growth and Fracture Toughness Testing, ASTM, STP 536 (1973) pp. 731–745.
Andren, R.C., Powell, T.J., Lawrence, F.V., Kesler, C.E., ‘Evaluation of the Fracture Behaviour of Concrete with the J-Integral’, Report NSF/ECS-81003, Department of Civil Engineering, University of Illinois at Urbana—Champaign, April 1981.
Halvorsen, G.T., ‘J-Integral Study of Steel Fibre Reinforced Concrete’,Int. J. Cem. Comp.,2 (1) (1980) 13–22.
Halvorsen, G.T., ‘Jm Toughness Comparison for some Plain Concretes’,Idem,2 (1980) 143–148.
Strawley, J.E., On the Relation of J to Work Done Per Unit Uncracked Area: ‘Total’ or Component ‘Due to Crack’,Int. J. Fracture 12 (1976) 470–474.
Buzzard, R.J., Fisher, D.M., ‘Load-Displacement Measurements and Work Determination in Three Point Bend Tests of Notched or Precracked Speciments’,Journal of Testing and Evaluation, JTEVA,6 (1) (1978) 35–39.
Keller, H.P., Munz, D., ‘Comparison of Different Equations for Calculation of J from One Load-Displacement Curve for Three-Point Bend Specimens’,Int. J. Fracture 12 (1976) 780–784.
American Society for Testing and Materials, ASTM E 813-81, ‘Standard Test Method for Jic, A Measure of Fracture Toughness’, Annual Book of Standards, 1986.
Rice, J.R., ‘A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks’, Transactions of the ASME,J. App. Mech. 35, Series E (2) (1968) 379–386.
Brown, W.F., Srawley, J.E., ‘Plane Strain Crack Toughness Testing of High Strength Metallic Materials’, ASTM Special Technical Publication, 410, 1966, pp. 84–86.
Srawley, J.E., ‘Wide Range Stress Intensity Factor Expressions for ASTM E-399 Standard Fracture Toughness Specimens’,Int. J. Fracture 12 (1976) 475–476.
American Society for Testing and Materials, ASTM E 399-83, ‘Standard Test Method for Plane Strain Fracture Toughness of Metallic Materials’, Annual Book of Standards, 1986.
Go, C.G., Swartz, S.E., Hu, K.K., ‘Stress Intensity Factors-for Single Edge Notch Beam’,J. Eng. Mech. ASCE 110 (6) (1984) 629–633.
Nallathambi, L., Karihaloo, B.L., Heaton, B.S., ‘Effect of Specimen and Crack Sizes, Water/Cement Ratio and Coarse Aggregate Texture upon Fracture Toughness of Concrete’,Magazine of Concrete Research 36 (129) (1984) 227–236.
Swartz, S.E., Fracture Mechanics: Applicability to Cracking and Fracture of Concrete, NCEL report CR-86.006, Kansas State University, 1986.
Wecharatana, M., Shah, S.P., ‘Predictions of Non Linear Fracture Process Zone in Concrete’, ASCE,J. Eng. Mech. 109 (5) (1983) 1231–1246.
Sih, G.C., ‘Non-linear Response of Concrete: Interaction of Size, Loading Step and Material Property’,Application of Fracture Mechanics to Cementitious Composites, NATO-ARW, September 4–7, 1984, Northwestern University (edited S.P. Shah) pp. 3–23.
Sih, G.C., ‘Mechanics of Material Damage in Concrete’,Fracture Mechanics of Concrete edited by Carpinteri, A. and Ingraffea, A.R. (Martinus Nijohff Publishers, Dordrecht, The Netherlands, 1984) pp. 1–29.
Shah, S.P., ‘Dependence of Concrete Fracture Toughness on Specimen Geometry and on Composition’,ibid, pp. 111–135.
Carpinteri, A., ‘Scale Effects in Fracture of Plain and Reinforced Concrete Structures’,Fracture Mechanics of Concrete edited by Sih, G. C., and DiTommaso, A. (Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1985) pp. 95–140.
Wecharatana, M., Shah, S. P., ‘Slow Crack Growth in Cement Composites’,ASCE Journal of the Structural Division 108 (ST6) (1982) 1400–1413.
Bazant, Z. P., ‘Instability, Ductility and Size Effect in Strain-Softening Concrete’J. Eng. Mech. Division, ASCE 102 (1976) 331–344.
Hildsorf, H. K., Brameshuber, W., ‘Size Effects in the Experimental Determination of Fracture Mechanics Parameters’, Application of Fracture Mechanics to Cementitious Composites, edited by S. P. Shah, NATO-ARW, September 4–7, 1984, Northwestern University, pp. 361–397.
Watson, K. L., ‘The Estimation of Fracture Surface Energy as a Measure of the Toughness of Hardened Cement Paste’,Cem. Concr. Res.,8 (1978) 651–656.
Cooper, G.A., ‘Optimization of the Three-Point Bend Test for Fracture Energy Measurement’,J. Mater. Sci. 12 (1977) 277–289.
Moavenzadeh, F., Kuguel, R., ‘Fracture of Concrete’,J. Mater. JMLSA 4 (3) (1969) 497–519.
Nakayama, J., Abe, H., Bradt, R.C., ‘Crack Stability in the Work of Fracture Test: Refractory Applications’,J. Amer. Ceram. Soc. 64 (11) (1981) 671–675.
Brown, E.T., Hudson, J.A., and Marathe, M.S., Discussions of the Paper: ‘Griffith Fracture Criterion and Concrete’,J. Eng. Mech. Division, ASCE,98 (EM5) (1972) 1310–1312.
Tattersall, H.G., Tappin, G., ‘The Work of Fracture and its Measurement in Metals, Ceramics and other Materials’,J. Mater. Sci. 1 (1966) 296–301.
Gurney, C., Hunt, J., ‘Quasi-Static Crack Propagation’,Proc. R. Soc., A299, 1967, 508–524.
Bazant, Z.P., ‘Mechanics of Fracture and Progressive Cracking in Concrete Structures’,Fracture Mechanics of Concrete edited by Sih, G.C. and DiTommasso, A. (Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1985) pp. 1–94.
Petersson, P.E., Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials, Report TVBM-1006, Division of Building Materials, Lund Institute of Technology, S-221 00 Lund, Sweden, 1981.
Hawkins, N.M., Wyss, A.N., Mattock, A.H., ‘Fracture Analysis of Cracking in Concrete Beams’,ASCE J. Struct. Division,103 (ST5) 1977, 1015–1030.
Hillerborg, A., ‘Numerical Methods to Stimulate Softening and Fracture of Concrete’, Fracture Mechanics of Concrete, edited by Sih G.C. and DiTommaso, A. (Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1985), pp. 141–170.
Li, V.C., ‘Fracture Resistance Parameters for Cementitious Materials and their Experimental Determinations’, Application of Fracture Mechanics to Cementitious Composites, NATO-ARW, September 4–7, 1984, Northwestern Univ., edited by S.P. Shah, pp. 431–449.
Reinhardt, H.W., ‘Crack Softening Zone in Plain Concrete under Static Loading’,Cem. Concr. Res.,15 (1985) 42–52.
Nakayama, J., ‘Bending Methods for Direct Measurement of Fracture Energy of Brittle Materials’,Jpn. App. Phys. 3 (7), (1964) 422–423.
Mai, Y.W., A Discussion of the Paper: ‘The Estimation of Fracture Surface Energy as a Measure of the Toughness of Hardened Cement Paste’,Cem. Concr. Res. 9 (1979) 537–539.
Watson, K.L., Reply to a Discussion of: ‘The Estimation of Fracture Surface Energy as a Measure of the Toughness of Hardened Cement Paste’,Cem. Concr. Res. 9 (1979) 591–599.
Go, C.G., Fracture Toughness Techniques to Predict Crack Growth and Tensile Failure in Concrete, PhD Dissertation, Kansas State University, 1983, p. 109.
Swartz, S.E., Yap, S.T., ‘Evaluation of Recently Proposed Recommendations for the Determination of Fracture Parameters for concrete in Bending’,Proc. VIII International Conference on Experimental Strain Analysis, Amsterdam, May 1986.
Swartz, S.E., Go, C.G. ‘Validity of Compliance Calibration to Cracked Concrete Beams in Bending’,Experimental Mechanics,24 (2) (1984) 129–134.
Swartz, S.E., Yap, S.T., ‘Evaluation of Proposed Methods to Determine Fracture Parameters for Concrete in Bending’, Research Report 181, Kansas State University, June 1986.
Mai, Y.W., ‘Fracture Measurements of Cementitious Composites’, Application of Fracture Mechanics to Cementitious Composites, NATO-ARW, September 4–7, 1984, Northwestern University, edited by S.P. Shah, pp. 399–429.
Jenq, Y. S., Shah, S.P., ‘Nonlinear Fracture Parameters for Cement Based Composites: Theory and Experiments’,ibid. Application of Fracture Mechanics to Cementitious Composites, NATO-ARW, September 4–7, 1984, Northwestern University, edited by S.P. Shah, pp. 319–359.
Walsh, P.F., ‘Crack Initiation in Plain Concrete’,Magazine of Concrete Research,28 (94) (1976) pp. 37–41.
Barr, B.I., Hasso, E.B.D., Sabir, B.B., ‘The Effect of Test Specimen Size on the Fracture Toughness of Concrete’,Cem. Concr. Res.,15 (1985) 833–841.
Jenq, Y., Shah, S.P., ‘Two Parameter Fracture Model for Concrete’,J. Eng. Mech., ASCE,111 (10) (1985) 1227–1261.
Visalvanitch, K., Naaman, A.E., ‘Fracture Methods in Cement Composites’,J. Eng. Mech. Division, ASCE,107 (EM5) (1981) 1155–1171.
Gjorv, O.E., Sorensen, S.I., Arnesen, A., A reply to the Discussion of the Paper: ‘Notch Sensitivity and Fracture Toughness of Concrete’,Cem. Concr. Res.,8 (1978) 389.
Belytschko, T., Bazant, Z.P., Hyun, Y.W., Chang, T.P., ‘Strain-Softening Materials and Finite-Element Solutions’,Comp. Struct. 23 (1986) 163–180.
Bazant, Z.P., Cedolin, L., ‘Blunt Crack Band Propagation in Finite Element Analysis’,J. Eng. Mech. Division, ASCE,105 (EM2) (1979) 297–315.
Strange, P.C., Bryant, A.H., ‘Experimental Tests on Concrete Fracture’,ibid,105 (EM2) (1979) 337–342.
Mindess, S., ‘The J-integral as a Fracture Criterion for Fiber Reinforced Concrete’,Cem. Concr. Res. 7 (1977) 731–742.
Gjorv, O.E., Sorensen, S.I., Arnesen, A., ‘Notch Sensitivity and Fracture Toughness of Concrete’,Cem. Concr. Res. 7 (1977) 333–344.
Shah, S.P., McGarry, F.J., ‘Griffith Fracture Criterion and Concrete’,J. Eng. Mech. Division, ASCE,97 (EM6) (1971) 1663–1676.
Higgins, D.D., Bailey, J.E., ‘Fracture Measurements on Cement Paste’,J. Mater. Sci. 11 (1976) 1995–2003.
Strange, P.C., Bryant, A.H., ‘The Role of Aggregate in the Fracture of Concrete’,ibid. 14 (1979) 1863–1868.
Carpinteri, A., ‘Static and Energetic Fracture Parameters of Rocks and Concrete’,Mat. Struct. 14 (81) (1981) 151–162.
Modeer, M.A., A Fracture Mechanics Approach to Failure Analysis of Concrete Materials, Report TVBM-1001, Division of Building Materials, Lund Institute of Technology, S-221 00 Lund, Sweden, 1979.
Hillerborg, A., Influence of Beam Size on Concrete Fracture Energy Determined According to a Draft RILEM Recommendation, Report TVBM-3021, Division of Building Materials, Lund Institute of Technology, S-221 Lund, Sweden, 1985.
Andrews, E.H., ‘A Generalized Theory of Fracture Mechanics’,J. Mater. Sci. 9 (1974) 887–894.
Andrews, E.H., Billington, E.W., ‘Generalized Fracture Mechanics, Part 2: Materials Subject to General Yielding’,J. Mater. Sci. 11 (1976) 1354–1361.
Andrews, E.H., Fukahori, Y., ‘Generalized Fracture Mechanics, Part 3: Prediction of Fracture Energies in Highly Extensible Solids’,J. Mater. Sci. 12 (1977) 1307–1319.
Irwin, G.R., Fracture Dynamics Fracturing of Metals, American Society of Metals, Cleveland, 1948, p. 152.
Orowan, E., Fundamentals of Brittle Behaviour in Metals, MIT Symposium on Fatigue and Fracture in Metals, Wiley, 1950, p. 139.
Maugis, D., ‘Subcritical Crack Growth, Surface Energy, Fracture Toughness, Stick-slip and Embrittlement’,J. Mater. Sci. 20 (1985) 3041–3093.
Orowan, E., ‘Fracture Strength of Solids’,Report on Progress in Physics 12, Physical Society of London, 1969.
Mindess, S., Diamond, S., ‘The Cracking and Fracture of Mortar’,Mater Struct.,15 (86) 1982, 107–112.
Mindess, S., ‘Rate of Loading Effects on the Fracture of Cementitious Materials’, Application of Fracture Mechanics to Cementitious Composites, edited by S.P. Shah, NATO-ARW, September 4–7, 1984, Northwestern University, pp. 617–636.
Petersson, P.E., Comments on the Method of Determining the Fracture Energy of Concrete by Means of Three-Point Bend Tests on Notched Beams. Report TVBM-3011, Division of Building Materials, Lund Institute of Technology, S-221 00 Lund, Sweden, 1982.
Mindess, S., ‘The Effect of Specimen Size on the Fracture Energy of Concrete’,Cem. Concr. Res. 14 (1984) 431–436.
Gillespie, R.L., Go, C.G., Swartz, S.E., ‘Vertical-Displacement Gage Fixture for Jic Testing’,Exper. Tech. 8 (5) (1984) 32–33.
Aluminum Company of America, ‘Alcoa Aluminium Handbook’, Alcoa Building, Pittsburgh, Pa.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Javier Malvar, L., Warren, G.E. Fracture energy for three point bend tests on single edge notched beams: Proposed evaluation. Materials and Structures 20, 440–447 (1987). https://doi.org/10.1007/BF02472495
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02472495